One of Fermat's theorems says that every prime number that yields a
remainder of 1 when divided by 4 can be expressed as the sum of two
integer squares (e.g.: 97 = 4^2 + 9^2). This theorem was proven by
Fermat. What methods are known for determining the two squares?

Find four distinct positive integers, a, b, c, and d, such that each
of the four sums a+b+c, a+b+d, a+c+d, and b+c+d is the square of an
integer. Show that infinitely many quadruples (a,b,c,d) with this
property can be created.

In a graph with infinite "points," if we colour the lines with two colors
we'll have either a red or a blue infinite chain of lines, an infinite
number of points, all of them joined to each other with the same
colour...

I have heard that Graham's number is the largest number with
mathematical use. I have seen it expressed in arrow notation but
that does not give me a sense of how large it is. Is there a way to
express the number of digits it contains?

Let X be a positive integer, A be the number of even digits in that
integer, B be the number of odd digits and C be the number of total
digits. We create the new integer ABC and then we apply that process
repeatedly. We will eventually get the number 123! How can we prove
that?

Given the equation 5y - 3x = 1, how can I find solution points where x
and y are both integers? Also, how can I show that there will always be
integer points (x,y) in ax + by = c if a, b and c are all integers?

Two positive integers are such that the difference of their squares is
a cube and the difference of their cubes is a square. Find the
smallest possible pair and a general solution for all pairs (a,b) that
satisfy the statement.